Mathematical Sciences: Analysis and Numerical Methods in Stochastic Optimization

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9529738 Yin This project in stochastic optimization consists of four parts: an examination of the properties of singularly perturbed Markov chains; a development of global stochastic optimization algorithms; a study of passive stochastic approximation procedures; and the design of numerical methods for stochastic optimization in manufacturing systems. The first part is intended to derive asymptotic properties of singularly perturbed nonstationary Markov chains and perturbed chains with weak and strong interactions, and to obtain asymptotic optimality in related control problems. The second part comprises the development of asymptotic properties of global stochastic optimization algorithms. By using weak convergence methods, convergence and rates of convergence are to be established, and the convergence of suitably scaled sequences to the solutions of appropriate stochastic differential equations is to be ascertained. In the third part, by combining stochastic approximation methods with kernel estimators, passive stochastic approximation algorithms are to be obtained. Here the goal is to establish convergence of the algorithms and to obtain error estimates. The fourth part encompasses numerical procedures for solving a class of robust control problems of piecewise deterministic Markov chains, and the development of stochastic gradient descent algorithms for approximating the optimal threshold values under the WIP (work in progress) policies for multi-machine manufacturing systems. Convergence and asymptotic properties of the algorithms are to be obtained. The entire research project will include two components: analysis and simulation. The main objectives are to enhance basic understanding of the asymptotic properties of the underlying systems and to develop sound and feasible algorithms. This proposal consists of four closely related topics. It is a bridge-building attempt to link theory and application in stochastic optimization. Emphasis is on the development of efficient numerical methods via analysis and numerical experimentation. The first topic deals with systems that are subject to rapid and random variations. The results obtained are intended be useful for applications involving hierarchical decision making, production planning, queuing networks in communication, and system reliability. To meet the increasing demand for efficient numerical procedures for global optimization, the second part of the project focuses on the development of numerical algorithms when random errors in the data have to be taken into account. The results are applicable to an ever expanding range of applications in estimation, identification and optimization problems. The research undertaken in the third part of this program derives from steady state estimation and detection problems in chemical engineering (for a continuously stirred tank and for a binary distillation column). The emphasis is on the design and implementation of numerical algorithms, which are expected to also have a variety of applications in target recognition, tracking, system failure detection, signal processing and related fields. The fourth part of the program is concerned with optimization methods for unreliable manufacturing systems. By emphasizing system stability rather than optimality alone, numerical methods for a class of robust control problems of production planning are to be developed. Under nowadays popular Kanban policies, which were originally promoted by the Japanese auto industry, a numerical procedure that will provide a systematic way of finding the optimal number of Kanbans for manufacturing systems is to be developed.. ***